Introduction to Partial Differential Equations

04/21/23http://

numericalmethods.eng.usf.edu 1

Introduction to Partial Introduction to Partial Differential EquationsDifferential Equations

http://numericalmethods.eng.usf.edu

Transforming Numerical Methods Education for STEM Undergraduates

For more details on this topic

Go to http://numericalmethods.eng.usf.edu

Click on KeywordClick on Introduction to Partial

Differential Equations

You are freeYou are free

to Share – to copy, distribute, display and perform the work

to Remix – to make derivative works

Under the following conditionsUnder the following conditionsAttribution — You must attribute the

work in the manner specified by the author or licensor (but not in any way that suggests that they endorse you or your use of the work).

Noncommercial — You may not use this work for commercial purposes.

Share Alike — If you alter, transform, or build upon this work, you may distribute the resulting work only under the same or similar license to this one.

What is a Partial What is a Partial Differential Equation ?Differential Equation ? Ordinary Differential Equations have only one independent

variable

Partial Differential Equations have more than one independent variable

subject to certain conditions: where u is the dependent variable, and x and y are the independent variables.

5)0(,353 2 yeydx

dy x

222

2

2

2

3 yxy

u

x

u

Example of an Ordinary Example of an Ordinary Differential EquationDifferential Equation

Assumption: Ball is a lumped system.Number of Independent variables:

One (t)

Hot Water

Spherical Ball

dt

dmChA a

Example of an Partial Example of an Partial Differential EquationDifferential Equation

Assumption: Ball is not a lumped system.Number of Independent variables: Four

(r,θ,φ,t)

Hot Water

Spherical Ball

aTrTtt

TC

T

r

kT

r

k

r

Tr

rr

k

)0,,,(,0,sin

sinsin 2

2

2222

2

Classification of 2Classification of 2ndnd Order Order Linear PDE’sLinear PDE’s

where are functions of ,and is a function of

02

22

2

2

Dy

uC

yx

uB

x

uA

CBA and,,yx and D

, , and , .u u

x y ux y

Classification of 2Classification of 2ndnd Order Order Linear PDE’sLinear PDE’s

can be: Elliptic Parabolic Hyperbolic

02

22

2

2

Dy

uC

yx

uB

x

uA

Classification of 2Classification of 2ndnd Order Order Linear PDE’s: Linear PDE’s: EllipticElliptic

02

22

2

2

Dy

uC

yx

uB

x

uA

042 ACBIf ,then equation is elliptic.

Classification of 2Classification of 2ndnd Order Order Linear PDE’s: Linear PDE’s: EllipticElliptic

02

22

2

2

Dy

uC

yx

uB

x

uA

Example:

where, giving

therefore the equation is elliptic.

02

2

2

2

y

T

x

T

1,0,1 CBA

04)1)(1(4042 ACB

Classification of 2Classification of 2ndnd Order Order Linear PDE’s: Linear PDE’s: ParabolicParabolic

02

22

2

2

Dy

uC

yx

uB

x

uA

2 4 0B AC If ,then the equation is parabolic.

Classification of 2Classification of 2ndnd Order Order Linear PDE’s: Linear PDE’s: ParabolicParabolic

02

22

2

2

Dy

uC

yx

uB

x

uA

Example:

where, giving

therefore the equation is parabolic.

2

2

x

Tk

t

T

0,0, CBkA

ACB 42 ))(0(40 k 0

Classification of 2Classification of 2ndnd Order Order Linear PDE’s: Linear PDE’s: HyperbolicHyperbolic

02

22

2

2

Dy

uC

yx

uB

x

uA

2 4 0B AC If ,then the equation is hyperbolic.

Classification of 2Classification of 2ndnd Order Order Linear PDE’s: Linear PDE’s: HyperbolicHyperbolic

02

22

2

2

Dy

uC

yx

uB

x

uA

Example:

where, giving

therefore the equation is hyperbolic.

2

2

22

2 1

t

y

cx

y

2

1,0,1

cCBA

)1

)(1(4042

2

cACB

0

42

c

THE ENDTHE ENDhttp://numericalmethods.eng.usf.edu

This instructional power point brought to you byNumerical Methods for STEM undergraduatehttp://numericalmethods.eng.usf.eduCommitted to bringing numerical methods to the undergraduate

AcknowledgementAcknowledgement

For instructional videos on other topics, go to

http://numericalmethods.eng.usf.edu/videos/

This material is based upon work supported by the National Science Foundation under Grant # 0717624. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

The End - ReallyThe End - Really